To the theory of semi-linear equations in the plane

• Veröffentlicht in: Український математичний вісник
• Datum: 2019
• Hauptverfasser: Gutlyanskii, V.Ya., Nesmelova, O.V., Ryazanov, V.I.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут прикладної математики і механіки НАН України 2019
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/169434
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: To the theory of semi-linear equations in the plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 105-140. — Бібліогр.: 74 назв. — англ.

In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C(Ď )∩ W¹,² loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semi-linear equations of mathematical physics, arising from modelling processes in anisotropic and inhomogeneous media. With a view to further development of the theory of boundary value problems for the semi-linear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.